Alphabeta MathHow statement and proof provenance work
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Exact valuation inputs for the topology comparison
PDF p. 6: Proposition 2.6 proof
Statement
For an étale algebra over a valuation ring , localization at a point over its closed point is a valuation-ring extension with the same value group. A proper morphism admits the valuative lift over whenever its fraction-field lift is given. Arc covers are tested using such extensions with valuation rank at most one.
External input
Import only these precise valuation/criterion facts, not an unrecorded comparison between finite-type and qcqs sites.
Facts & Assumptions
Given: The hypotheses in the statement.
Goal: Establish the stated exact map or descent conclusion.
The exact source records in External input.
Proof
Import exactly the two Stacks assertions and the BM definition in the source records.
In the application the proper morphism is finitely presented and separated, so the criterion’s finite-type and quasi-separated hypotheses hold. Equality of value groups in the étale assertion preserves rank. These are exact foundational inputs; the generator comparison itself is authored locally.
Depends on
Dependency tree · 1 level
Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.
External dependencies
If A is a valuation ring, A→B is étale, and m lies over the maximal ideal, then A⊂B_m is an extension of valuation rings with identical value groups.
Hypotheses: Arbitrary valuation ring and étale ring map, no noetherian hypothesis.
Reading limits: Read full lemma and displayed proof; the deeper weak-dimension and normalization results cited there are imported, not reconstructed.
A proper finite-type quasi-separated morphism has a unique lift over any valuation-ring diagram with its fraction field.
Hypotheses: Morphisms of schemes; arbitrary valuation ring; the proper maps here are finitely presented, hence finite type, and separated, hence quasi-separated.
Reading limits: Read full displayed statement and short proof; underlying universal-closedness/separatedness criteria not reconstructed.
Arc covers are tested by extensions of valuation rings of rank at most one.
Hypotheses: Qcqs source/target schemes and families with the source finite-subcover convention.
Reading limits: Read full definition and Proposition2.1 proof; the local generator lemma applies the rank-at-most-one definition directly.
Sources
- The Stacks Project, Extensions of valuation rings · Lemma15.125.5 and its full statement/proof
- The Stacks Project, Valuative criterion for properness · Lemma29.43.1, statement and proof
- Bhargav Bhatt and Akhil Mathew, The arc-topology · Definition1.2 p.3; Proposition2.1 and proof p.9